Pre-release campaigns rarely come as enticing as that preceding the Civil Wars' eponymous second album. In November, the duo announced they were splitting up, due to "internal discord and irreconcilable differences of ambition". People acted surprised: the band were generally held to be on the cusp of huge global success. There was also the weird on stage chemistry that ensured John Paul White and Joy Williams continually had to deny they were romantically involved: one LA Times profile noted White tenderly brushing a stray strand of hair from Williams' face as they duetted on-stage. But maybe the news of the split shouldn't have been so shocking. Thrown together at a Nashville songwriters' workshop, they had always seemed an unlikely pairing – Williams formerly a winsome Christian pop singer beloved of websites called things like jesusfreakhideout.com; White a mainstream alt-rocker who had perhaps learned from the White Stripes not just the power of a uniform, but also the value of a bit of are-they-aren't-they intrigue. Still, the suddenness of their exit was striking: the pair had apparently become so sick of the sight of each other that they were incapable of even completing a UK tour together. The announcement came swiftly after they left the stage at London's Roundhouse, and involved the cancellation of four further shows.

Then they appeared to suggest they were back in the studio to complete their second album. They had, it transpired, also invited a film crew along to document the sessions, which suggested a band keen to learn valuable lessons from rock history – the presence of Michael Lindsay Hogg's cameras having famously done so much to quell the bloodcurdling acrimony between the Beatles during the making of Let It Be. The resulting film about the making of The Civil Wars appears to be a study in white-lipped, ask-your-mother-to-pass-the-sugar froideur: uncomfortable silences and shots of the pair studiously avoiding each other's gaze, interspersed with Williams gamely noting that "great art is made from tension".

Of course, she's right. It's not just a prurient desire to hear the airing of dirty laundry set to music that makes all this a hugely appealing sell. From Rumours to Sister Ray, the past is packed with evidence that bands on the verge of throttling each other make compelling music. Besides, perhaps the Civil Wars could do with a bit more tension: if you were going to level a complaint at their debut album, Barton Hollow, it might have been that, for all its sparse loveliness, there were moments when its slickness brought it too close to the middle of the road.

A tougher, darker take on their debut is just what The Civil Wars' opening tracks suggest. Even if you're disinclined to read anything about the state of Williams and White's relationship into the lyrics – which takes some doing when you're confronted with the sound of them wailing "I wish I'd never seen your face" in unison – the tone is grippingly ominous. The One That Got Away and I Had Me A Girl set their tales of sickly relationships to frustrated-sounding explosions of slide guitar and serrated distortion. Even on Same Old Same Old, which veers closer to something Steve Wright might play after furnishing the nation with a hilarious factoid or two, there's an almost suffocating intensity about the pair's vocals, at odds with the musical backing.

There are other moments like these on The Civil Wars. Devil's Backbone takes a bunch of country cliches – a man on the run with mouths to feed, a penitent wife, much invoking of the good Lord – and delivers them with a ferocity that makes them compelling; Oh Henry would be a fantastic song however it was arranged, but the churning guitar noise underneath the vocals adds an intriguing hint of menace, while From This Valley features an astonishing a cappella section, the pair's voices stretching vowels and bending notes in perfect unison. Equally, however, there are moments when it heads in the opposite direction. If anything, Dust to Dust and Eavesdrop seem slicker than the debut, offering the kind of emotive rock ballad that gets played at the end of US TV dramas, as the action cuts between someone dying in hospital and a previously estranged couple kissing in the rain. The vocals are invariably great, but there are moments where you wonder if you're merely projecting a certain intensity on to them because you know the album's backstory.

In fairness, these are the kind of songs that might catapult the album to multi-platinum success – it doesn't take a huge leap of imagination to picture them racing up the charts in the wake of a terrifying assault at the hands of an X Factor semi-finalist – and it's worth noting that the Civil Wars deal in a superior brand of MOR, replete with beautiful melodic flourishes. But their second album makes it clear they can also do something more substantial. As it flips between its two poles, you can't help wondering whether the aesthetic gulf between them tells you more about "irreconcilable differences of ambition" than all the intense vocals and lyrics about dying relationships put together.